Geometric variational inference

TL;DR

This paper introduces geometric Variational Inference (geoVI), a novel method leveraging Riemannian geometry and Fisher information to improve approximation accuracy in high-dimensional, non-linear probability distributions.

Contribution

It proposes geoVI, a new variational inference approach that incorporates geometric information to enhance efficiency and accuracy in complex probabilistic models.

Findings

geoVI achieves accurate normal approximations in high-dimensional problems

The method demonstrates efficiency on hierarchical Bayesian inverse problems

GeoVI outperforms traditional VI methods in several examples

Abstract

Efficiently accessing the information contained in non-linear and high dimensional probability distributions remains a core challenge in modern statistics. Traditionally, estimators that go beyond point estimates are either categorized as Variational Inference (VI) or Markov-Chain Monte-Carlo (MCMC) techniques. While MCMC methods that utilize the geometric properties of continuous probability distributions to increase their efficiency have been proposed, VI methods rarely use the geometry. This work aims to fill this gap and proposes geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric. It is used to construct a coordinate transformation that relates the Riemannian manifold associated with the metric to Euclidean space. The distribution, expressed in the coordinate system induced by the transformation, takes a particularly simple form that allows for an accurate variational approximation by a normal distribution. Furthermore, the algorithmic structure allows for an efficient implementation of geoVI which is demonstrated on multiple examples, ranging from low-dimensional illustrative ones to non-linear, hierarchical Bayesian inverse problems in thousands of dimensions.